Abstract
In this paper, general beam vibration problems with several attachments under arbitrarily distributed harmonic loading are solved. A multiple-stepped beam is modelled by the Euler-Bernoulli beam theory and an extension of an efficient numerical method called Numerical Assembly Technique (NAT) is used to calculate the steady-state harmonic response of the beam to an arbitrarily distributed force or moment loading. All classical boundary conditions are considered and several types of concentrated elements (springs, dampers, lumped masses and rotatory inertias) are included. Analytical solutions for point forces and moments and polynomially distributed loads are presented. The Fourier extension method is used to approximate generally distributed loads, which is very efficient for non-periodic loadings, since the method is not suffering from the Gibbs phenomenon compared to a Fourier series expansion. The Numerical Assembly Technique is extended to include distributed external loadings and a modified formulation of the solution functions is used to enhance the stability of the method at higher frequencies. The method can take distributed loads into account without the need for a modal expansion of the load, which increases the computational efficiency. A numerical example shows the efficiency and accuracy of the proposed method in comparison to the Finite Element Method.
Highlights
In engineering applications, real structures can be frequently modelled by an assembly of beams with attached concentrated elements like lumped masses, springs and dampers, e.g. lever arms, shafts, aeronautical structures, robotic arms etc
The numerical example shows the overall excellent agreement of the results calculated with Numerical Assembly Technique (NAT) and Finite Element Method (FEM)
The results of NAT are expected to be highly accurate, since NAT is quasi-analytical in the sense that analytical homogeneous solutions and analytical particular solutions for point and polynomially distributed excitations are used
Summary
Real structures can be frequently modelled by an assembly of beams with attached concentrated elements like lumped masses, springs and dampers, e.g. lever arms, shafts, aeronautical structures, robotic arms etc. Compared to GFM, the GFA leads to a 2 × 2 system matrix for the twodimensional beam vibration problem independent of the number of concentrated elements and changes in the material parameters or cross-section. The Numerical Assembly Technique is extended to allow for the analysis of general beam vibration problems with any number of concentrated attachments (translational and rotational springs, translational and rotational dampers, lumped masses and rotatory inertia) excited by an arbitrarily distributed harmonic loading. Compared to the standard NAT, a different solution of the homogeneous governing equation is applied, which leads to a better conditioning of the system matrix, especially for higher frequencies and long and thin beams This enhances the stability and accuracy of the method. A small approximation error occurs if the external loading is not concentrated or polynomially distributed, but the results can be made arbitrarily accurate by increasing the order of the Fourier extension when approximating the generally distributed load
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